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Chordal quadrileteral problem

Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:

Attachment 27586"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90^{o}- 1/2*arc ED.

Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90^{o}- 1/2*arc ED.

=> Angle A = Angle EFC.

Angle A + Angle EFB = Angle EFC + Angle EFB = 180^{o}.

Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Thanks in advance,

Tom

Re: Chordal quadrileteral problem

Quote:

Originally Posted by

**tomkoolen** Hello everyone,

I'm having a bit of trouble understanding the following proof of this problem:

Attachment 27586"Prove that ABFE is a chordal quadrileteral."

Proof given by solutions book:

"Angle A = 1/2*arc CD - 1/2*arc ED = 90

^{o}- 1/2*arc ED.

Angle EFC = 1/2*arc EC = 1/2*(arc CD - arc ED) = 90

^{o}- 1/2*arc ED.

=> Angle A = Angle EFC.

Angle A + Angle EFB = Angle EFC + Angle EFB = 180

^{o}.

Thus ABFE is a chordal quadrileteral."

I do understand the logic of the proof and the whole conclusion, however I am stuck at the beginning, where the angles are given as arcs. Could anybody explain to me how this is done?

Thanks in advance,

Tom

Please define *chordal quadrileteral*.

Re: Chordal quadrileteral problem

It is in America more often called cyclic quadrilateral; a quadrilateral with all of its vertices on a circle.

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Re: Chordal quadrileteral problem

Hi,

In a circle an arc angle is the length of the arc divided by the radius of the circle (formula $\displaystyle s=r\theta$). Equivalently, an arc angle is the measure of the "central" angle; i.e. the angle from the center of the circle with endpoints the endpoints of the arc. Here's the theorem which is the first line of your proof:

Attachment 27692